Answer to Question #194732 in Statistics and Probability for james

Question

ILAW Manufacturing company produces bulbs that last a mean of 900 hours with a standard deviation of 110 hours. what is the probability that the mean lifetime of a random sample of 15 of these bulbs is less than 850 hours?

Answer 1

Let $X$ be the random variable which represents the length of the life of a light bulb, in hours: $X\sim N(\mu, \sigma^2/n)$

Then

$$Z={X-\mu \over \sigma/\sqrt{n}}\sim N(0,1)$$

Given that $\mu=900\ hrs, \sigma=110\ hrs, n=15.$

The probability that the mean lifetime of random sample of 15 of these bulbs is less than 850 hours is

$$P(X<850)=P(Z<{850-900 \over 110/\sqrt{15}})\approx P(Z<-1.3176)\approx$$ $$\approx0.0938$$